“Our test is applicable when the variance (over data samples) of the log probability ratio between the proposal and the current state is less than one.”

By test, the authors mean a mini-batch formulation of the Metropolis-Hastings acceptance ratio in the (special) setting of iid data. First they use Barker’s version of the acceptance probability instead of Metropolis’. Second, they use a Gaussian approximation to the distribution of the logarithm of the Metropolis ratio for the minibatch, while the Barker acceptance step corresponds to comparing a logistic perturbation of the logarithm of the Metropolis ratio against zero. Which amounts to compare the logarithm of the Metropolis ratio for the minibatch, perturbed by a logistic minus Normal variate. (The cancellation of the Normal in eqn (13) is a form of fiducial fallacy, where the Normal variate has two different meanings. In other words, the difference of two Normal variates is not equal to zero.) However, the next step escapes me as the authors seek to optimise the distribution of this logistic minus Normal variate. Which I thought was uniquely defined as such a difference. Another constraint is that the estimated variance of the log-likelihood ratio gets below one. (Why one?) The argument is that the average of the individual log-likelihoods is approximately Normal by virtue of the Central Limit Theorem. Even when randomised. While the illustrations on a Gaussian mixture and on a logistic regression demonstrate huge gains in computational time, it is unclear to me to which amount one can trust the approximation for a given model and sample size…

Scott Schmidler and his Ph.D. student Douglas VanDerwerken have arXived a paper on parallel MCMC the very day I left for Chamonix, prior to MCMSki IV, so it is no wonder I missed it at the time. This work is somewhat in the spirit of the parallel papers Scott et al.’s consensus Bayes, Neiswanger et al.’s embarrassingly parallel MCMC, Wang and Dunson’s Weierstrassed MCMC (and even White et al.’s parallel ABC), namely that the computation of the likelihood can be broken into batches and MCMC run over those batches independently. In their short survey of previous works on parallelization, VanDerwerken and Schmidler overlooked our neat (!) JCGS Rao-Blackwellisation with Pierre Jacob and Murray Smith, maybe because it sounds more like post-processing than genuine parallelization (in that it does not speed up the convergence of the chain but rather improves the Monte Carlo usages one can make of this chain), maybe because they did not know of it.

“This approach has two shortcomings: first, it requires a number of independent simulations, and thus processors, equal to the size of the partition; this may grow exponentially in dim(Θ). Second, the rejection often needed for the restriction doesn’t permit easy evaluation of transition kernel densities, required below. In addition, estimating the relative weights wi with which they should be combined requires care.” (p.3)

The idea of the authors is to replace an exploration of the whole space operated via a single Markov chain (or by parallel chains acting independently which all have to “converge”) with parallel and independent explorations of parts of the space by separate Markov chains. “Small is beautiful”: it takes a shorter while to explore each set of the partition, hence to converge, and, more importantly, each chain can work in parallel to the others. More specifically, given a partition of the space, into sets Ai with posterior weights wi, parallel chains are associated with targets equal to the original target restricted to those Ai‘s. This is therefore an MCMC version of partitioned sampling. With regard to the shortcomings listed in the quote above, the authors consider that there does not need to be a bijection between the partition sets and the chains, in that a chain can move across partitions and thus contribute to several integral evaluations simultaneously. I am a bit worried about this argument since it amounts to getting a random number of simulations within each partition set Ai. In my (maybe biased) perception of partitioned sampling, this sounds somewhat counter-productive, as it increases the variance of the overall estimator. (Of course, not restricting a chain to a given partition set Ai has the incentive of avoiding a possibly massive amount of rejection steps. It is however unclear (a) whether or not it impacts ergodicity (it all depends on the way the chain is constructed, i.e. against which target(s)…) as it could lead to an over-representation of some boundaries and (b) whether or not it improves the overall convergence properties of the chain(s).)

“The approach presented here represents a solution to this problem which can completely remove the waiting times for crossing between modes, leaving only the relatively short within-mode equilibration times.” (p.4)

A more delicate issue with the partitioned MCMC approach (in my opinion!) stands with the partitioning. Indeed, in a complex and high-dimension model, the construction of the appropriate partition is a challenge in itself as we often have no prior idea where the modal areas are. Waiting for a correct exploration of the modes is indeed faster than waiting for crossing between modes, provided all modes are represented and the chain for each partition set Ai has enough energy to explore this set. It actually sounds (slightly?) unlikely that a target with huge gaps between modes will see a considerable improvement from the partioned version when the partition sets Ai are selected on the go, because some of the boundaries between the partition sets may be hard to reach with a off-the-shelf proposal. (Obviously, the second part of the method on the adaptive construction of partitions is yet in the writing and I am looking forward its aXival!)

Furthermore, as noted by Pierre Jacob (of Statisfaction fame!), the adaptive construction of the partition has a lot in common with Wang-Landau schemes. Which goal is to produce a flat histogram proposal from the current exploration of the state space. Connections with Atchadé’s and Liu’s (2010, Statistical Sinica) extension of the original Wang-Landau algorithm could have been spelled out. Esp. as the Voronoï tessellation construct seems quite innovative in this respect.